The cyclotron is a device used to accelerate charged particles to high velocity. In the cyclotron, charged particles are confined to circular orbits by a magnetic field, and are accelerated with an electric field created by two hollow conductive semicircular electrodes termed dees. The frequency of the oscillating electrical potential across the dees is timed to be equal to the orbital frequency of the charged particle in the magnetic field, and it is constant as the energy of the particle increases with every turn. The particles in the cyclotron experience a resonant energy-multiplying effect as they always traverse the gap between the dees when the oscillatory electric field is at its maximum.
Problems arise when particles in the cyclotron attain speeds that are an appreciable fraction of the speed of light, and, according to the theory of special relativity, gain mass proportional to their kinetic energy. The increased mass causes the particles' resonant cyclotron frequency to change, and they fall out of step with the applied electric field. Thus, as discussed in H. A. Bethe and M. E. Rose. 1937, The Maximum Energy Obtainable from the Cyclotron, Phys. Rev. 52: 1254-1255, the traditional cyclotron is fundamentally limited in the energy it can attain due to its fixed frequency and magnetic field.
Existing methods of overcoming relativistic mass increase and resonant frequency shift in the cyclotron also introduce severe limitations. As discussed in U.S. Pat. No. 2,615,129, and D. Bohm and L. L. Foldy. 1947. Theory of the Synchro-Cyclotron, Phys. Rev. 72: 649-661, Synchrocyclotrons modulate the cyclotron frequency with time, to keep a single packet of particles synchronous throughout the accelerator at a time. Synchrocyclotrons can accelerate to higher energies, but they yield only low-intensity beams of particles, as the duty cycle of the accelerated beam is small. Isochronous cyclotrons continue to drive relativistic particles with a constant frequency by increasing magnetic field strength with increasing radius, as the cyclotron frequency is given by:
  f  =      qB          2      ⁢      π      ⁢                          ⁢      M      where B is magnetic field strength, q is charge of the particle, and M is the mass of the particle.
However, isochronous cyclotrons are also limited in that for the most efficient use of space, and the most cost-efficient way to attain high-energy particles, the magnetic field would be as high as possible throughout the acceleration area. With conventional resistive electromagnets, an iron yoke saturates at about 2 Tesla, and the magnet cannot be operated at a higher field; to attain higher-energy particles would require increasing the magnet's radius while still keeping the correct gradient, resulting in a smaller magnetic field in the center. Further, weak-focusing cyclotrons cannot be isochronous, and therefore, more complicated beam focusing techniques must be used in isochronous cyclotrons.